Quantum information

as the information of infinite series

Vasil Penchev • Dept. of Logical Systems and Models Institute for the Study of Societies and Knowledge Bulgarian Academy of Sciences vasildinev@gmail.com _______________________ The conference “Quantum Computation, Quantum Information, and the Exact Sciences” Ludwig Maximilian University of Munich 30-31 January 2015, 14:15-15:00 (30th, Friday)

1 INTRODUCTION

The thesis:

Quantum information (introduced by quantum mechanics) is equivalent to that generalization of the classical information from finite to infinite series or collections

All physical processes turn out to be quantum-informational, and nature or the universe is a quantum computer processing quantum information

The main statement:

Quantum information is equivalent to the generalization of information from finite to infinite series

• The axiom of choice is necessary for quantum information in two ways:

• (1) in order to guarantee the choice even if any constructive approach to be chosen an element of the continuum does not exist

• (2) to equate the definition in terms of Hilbert space and that as a choice among a continuum of alternatives:

• Indeed the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1968) demonstrate that the mathematical formalism of quantum mechanics implies that no well-ordering of any coherent state might exist before measurement

• However, the same coherent state is transformed into a well-ordered series of results in time after measurement

• In order to be equated the state before and after measurement, the well-ordering theorem equivalent to the axiom of choice is necessary

• The measurement is an absolutely random choice of an element of the coherent state, for which no constructive way can exist in principle

• The quantity of quantum information can describe uniformly the state before and after measurement (equivalent to a choice among an infinite set)

• Hilbert space can be understood as the free variable of quantum information

• Then any wave function, being a given value of it, “bounds” both unorderable and a well-ordered state as the quantity of qubits (i.e. the “infinite choices”) necessary for the latter to be obtained from the former

• Hilbert space can be understood as the free variable of quantum information

• Then any wave function, being a given value of it, “bounds” both unorderable and a well-ordered state as the quantity of qubits (i.e. the “infinite choices”) necessary for the latter to be obtained from the former

• The quantity of quantum information is the ordinal corresponding to the infinity series in question

• The ordinal number corresponds one-to-one to a coherent state as the one and same quantity of quantum information containing in it

2 THE NOTIONS AND GENERAL VIEWPOINT

• The generalization of information, to which quantum mechanics is forced in order to be able to be reformulated in terms of information, is necessary for the solution of its basic problem:

• How to be unified and then uniformly described quantum leaps, i.e. discrete morphisms, and classical motions, i.e. smooth morphism differentiating from each other by the availability of velocity in any point of the trajectory in the latter case?

• The solution turns out to involve necessarily infinity for it is impossible for any finite mathematical structure

• So, the complex Hilbert space, which just underlies quantum mechanics, is not only infinitely dimensional in general, but furthermore just being complex, it requires a continuum to be defined even for any finite set of dimensions

Quantum mechanics turns out to be the first and probably single experimental science utilizing infinity necessarily therefore being in condition to make experiments on infinity in final analysis

• If quantum information refers to infinity in the sense above, the Kolmogorov definition seems to be to be generalized both to transfinite sequences and to series of transfinite ordinal numbers

• However, there is not an obvious way for that meaningful generalization at first glance

• The main statement of this paper will attempt to resolve the problem about that generalization

• It is worth to add that the Kolmogorov definition interprets the concept of order as a well-ordering in the rigorous mathematical sense of the latter term

“Choice” • Though both information and quantum information

can be understood and defined as a relevant mathematical relation of orders, they can be equivalently represented as the quantities of elementary choices, correspondingly bits and qubits

• “Choice” turns out to be more rigorous and precisely determinable concept and thus more relevant as the ground of information than “order”

“Choice as the “atom” of the course of time”

• “Choice” can be defined absolutely independently of “time”

• However it can also be interpreted as an “atom“ of time unifying the minimally possible “amount” of future, present, and past

• For example, the future is symbolized by the “empty cell” of the choice, the present by the action itself of choice, and the past by the chosen alternatives

• One can add that the choice as to the physical course of time is “infinite” for quantum information measurable by units of “infinite choice” is what underlies nature

• The choice from finiteness to infinity can be generalized at the expense of the loss of constructiveness, which loss in turn can be interpreted as “randomness”

• One can demonstrate that quantum mechanics needs the axiom of choice:

• Indeed the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1968) demonstrate that the mathematical formalism of quantum mechanics implies that no well-ordering of any coherent state might exist before measurement

The complex “Hilbert space” 1. It synthesizes arithmetic and geometry, positive integers and Euclidean space 2. Its elements resolve the problem: how to be united quantum leaps and smooth trajectories 3. Its elements can be interpreted both as future states and as actual states of some quantum system 4. It coincides with the space dual to it: Thus the elements of the dual space are identical but complement to those of the initial Hilbert space 5. Though the description by Hilbert space needs only the “half” of variables in comparison to classical mechanics, any “hidden variables” cannot be added in principle

“Quantum mechanics as a theory of quantum information”

• Quantum mechanics can be exhaustedly rewritten in the language of quantum information. This means that the base of nature is entirely information:

• There is nothing in the world besides quantum information!

3 THE PROOF AND MOST COMMENTS ON THE MAIN STATEMENT

The first remark • Hilbert space is intrinsically invariant to the axiom

of choice • This can be demonstrated by means of its

interpretation in terms of quantum mechanics • Any element of it describes equally well both state

of a statistical ensemble therefore well-ordered and corresponding coherent state “by itself” (i.e. before measurement) therefore unorderable always and in principle

• The validity of those both implies the well-ordering theorem equivalent to the axiom of choice while only the latter state excludes it

• Thus the combination of both in turn implies the invariance to the axiom of choice as to any element of Hilbert space therefore interpretable both as a transfinite well-ordering and as the quantity of the elements of the set “before that well-ordering” (and even excluding it in principle)

The third remark • Any “wave function” indicates unambiguously just

one transfinite series therefore being able to serve both as its name and as the “quantity of infinity” unlike the mere infinity as quality

• Any way for the transfinite series to be named unambiguously is necessarily constructive and exhaustedly describable by means of its name, i.e. by the point of Hilbert space in the case in question

• However, any indication about the unambiguous well-ordering of them is not possible just because of the duality to the axiom of choice and thus to the well-ordering

Much to many

• One can utilize the metaphor of ordering a “much” into some corresponding “many”

• Any ordering will be constructive for the results (any given “many” will be a result) are able to be one-to-one mapped into the set of all those orderings as they cannot depend on that “much” in principle just in virtue of its nature to be merely “much”

A Rorschach spot • It can be seen so or otherwise, e.g. as “two horses”

or as “a single butterfly” • Each seeing as a given “something” is constructive

as the seeing of the something is determined only by the seeing rather than by the Rorschach spot itself

• The random naming even of the same something is not constructive being irreproducible in principle

• Analogically, that choice guaranteed by the axiom of choice is random, irreproducible and unconstructive in principle: “Seeing” or “naming” the same selected element being already reproducible is constructive

4 INTERPRETATIONS

• The main statement above means just this: Hilbert space and all “points” in it can be assigned to infinity as a set of discerning “names” for all infinite sets

• This is a method for the doctrine of infinity in mathematics to be transformed from a rather qualitative into a rigorous quantitative theory directly applicable in a series of areas

Hilbert space a generalization of Peano’s arithmetic

• Hilbert space is a free variable of quantum information as a well-ordered series of empty qubits

• Any point in it (i.e. any “wave function”) is a value of it therefore transforming it into a bound variable

• The Nth qubit can be interpreted as a “geometrical” generalization of the positive integer N after substituting the point “N” with the unit ball “N”

• Thus all values of this qubit “N” corresponds to the single point “N” after the “arithmetical degeneration” collapsing the qubit into a point

A “mathematical metaphor”

• The main theorem of algebra finds the relevant set of numbers, which turns out to be the field of complex numbers, so that any equation with coefficients from this set to possesses solutions, which are elements only of the same set

• The main theorem of algebra in order to elucidates the way for the main statement of this paper to be referred to the foundation of mathematics:

The “scholia” of the metaphor • One should search for that set, which is relevant

to completeness for the granted one (e.g. that of real numbers in the case of the main theorem of algebra) can turn out to be unsufficient

• Thus, arithmetic granted to be the necessary and self-obvious base for set theory and via it, to all mathematics is unsufficient as the Gödel theorems demonstrate

• One can search for that set, possibly including arithmetic, which is indeed relevant, independently of the prejudice

Right the complex Hilbert space by its properties both to unify geometry and arithmetic and to name infinity in detail is a relevant applicant for the base of the cherished completeness and therefore, of the self-foundation of all mathematics

A few comments 3. Summarizing the above two considerations, one can conclude that the main statement interpreted in terms of information theory means generalizing the concept of information also to the past extending the invariance of description of entropy and expectation as to well-ordered number series • One can simply say: Information is invariant to

time, to past, present, and future, therefore being one of the most fundamental quantities both physical and mathematical

The Schrödinger equation is a generalization of the fundamental law of energy conservation in classical mechanics: • Indeed the former implies the latter if the

wave function is a constant as a particular case

From “The Schrödinger equation” in Wikipedia

A few comments: The potential and kinetic energy are clearly distinguished from each other in classical mechanics: • Kinetic energy corresponds to the real motion

depending only on its velocity, and potential energy describes the force in each spatial point acting on a tentative reference unit (e.g. a body or a material point) depending only on its position

• The law of energy conservation means the way, in which the acting of the force is transformed into the motion of some real item

• The Schrödinger equation generalizes the classical understanding as follows:

• The wave function corresponds to space in a different way in the cases of kinetic and potential energy:

• The tensor product of two identical and complementary (or “conjugate”) wave functions is mapped into space if the energy is kinetic, and a single wave function is mapped into space in the case of potential energy

• Indeed the latter can be interpreted as the action on an energy-momentum unit thus depending only on the spatial coordinates

• Summarizing all consideration above, one can suggest the following meaning of the Schrödinger equation in terms of quantum information:

• It generalizes energy conservation to past, present, and future moments of time rather than only to present and future moments as this does the analogical law in classical mechanics

• Thus, it is the universal law of how “time flows” or in other words, about the course of time

• Furthermore, resolving its problem, the Schrödinger equation suggests the proportionality (or even equality if the units are relevantly chosen) of the quantities of both quantum and classical information and energy therefore being a (quantum) information simile of Einstein’s famous equality of mass and energy (“E=mc2”)

5 CONCLUSIONS & FUTURE WORK

• All this implies some kind of neo-Pythagorean ontology making related mathematics, physics, and technics immediately, by an explicit mathematical structure

• Number and being (by the meditation of time), the natural and artificial turn out to be not more than different hypostases of a single common essence

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